Rank Enumeration and Side-Count Invariance for Noncrossing Hull Partitions
Overview
Take finitely many points on a convex polygon's boundary and partition them into blocks with pairwise disjoint convex hulls. The rank is the number of points minus the number of blocks. This paper counts the partitions at every rank and proves that permuting the numbers of internal points on the sides leaves all these counts unchanged. If every side has an internal point, the counts also determine the multiset of side counts. The motivating polygonal non-rank-symmetry conjecture of Dougherty and Root, and its reciprocity identity, were previously proved by Shivam Patel; the contribution here is the full enumeration and permutation invariance. Collinear configurations are excluded. The paper contains complete proofs, not yet formalized.
Original public questions
Dougherty–Root hull-configuration conjecture: full rank enumeration beyond the known polygonal solution
Michael Dougherty and Gina Root: Noncrossing Partitions From Hull Configurations — Introduction, unnumbered conjecture after Theorem A; polygonal restriction. The original non-rank-symmetry conclusion is prior work, not claimed as a new solution here.
Shivam Patel: Complete proof for every polygonal hull configuration, with an exact rank-reciprocity formula — August 20, 2026 solution; credited polygonal non-rank-symmetry proof and rank-reciprocity identity.
Original abstract (English)
Let P be a finite set on the boundary of its convex polygon, with c_i points in the relative interior of side i. We give an explicit formula for the rank polynomial of the partitions of P whose block convex hulls are pairwise disjoint. The formula is a linear evaluation of a product of explicitly defined side polynomials. In particular, it depends only on the multiset of the c_i, not on their cyclic order. Among configurations for which every side has an internal point, equal rank polynomials are equivalent to equal multisets of side counts. The same decomposition gives a direct proof of Patel's rank-reciprocity identity, together with directional rank inequalities and an exact mean-rank formula. The proof groups inclusion-exclusion terms into boundary tilings and separates local forbidden successor arcs from the single cycle that can wind around the entire boundary.
Mathematical review
The manuscript has completed internal review. Complete formalization is not yet established for this public version.
Review standard